# Stable Betti numbers of (partial) toroidal compactifications of the moduli space of abelian varieties

## TL;DR

This paper introduces an algorithm to compute stable cohomology and Betti numbers of certain toroidal compactifications of the moduli space of abelian varieties, linking combinatorics of fans with algebraic topology.

## Contribution

It provides a new explicit algorithm for calculating stable cohomology generators for rationally smooth partial toroidal compactifications, including the matroidal and perfect cone cases.

## Key findings

- Algorithm determines stable cohomology in terms of fan combinatorics.
- Computes stable Betti numbers near top degree for the perfect cone compactification.
- Suggests an algebra structure on the stable cohomology in near top degree.

## Abstract

We present an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of ${\mathcal A}_g$, satisfying certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular the algorithm determines the stable cohomology of the matroidal partial compactification, in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers in close to top degree for the perfect cone toroidal compactification. This suggests the existence of an algebra structure on the stable cohomology of the perfect cone compactification in close to top degree.